Problem: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$2.50$ each for students, and the group paid $$50.50$ in total. The next month, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$11.50$ each for students, and the group paid $$159.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+2.5y = 50.5}$ ${14x+11.5y = 159.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-5y = -101}$ ${14x+11.5y = 159.5}$ Add the top and bottom equations together. $ 6.5y = 58.5 $ $ y = \dfrac{58.5}{6.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7x+2.5y = 50.5}$ to find $x$ ${7x + 2.5}{(9)}{= 50.5}$ $7x+22.5 = 50.5$ $7x = 28$ $x = \dfrac{28}{7}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {14x+11.5y = 159.5}$ and get the same answer for $x$ ${14x + 11.5}{(9)}{= 159.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.